A Calabi–Yau threefold is called of type K if it admits an ´etale Galois covering by the product of a K3 surface and an elliptic curve. In our previous paper [16], based on Oguiso–Sakurai’s fundamental work [24], we have provided the full classification of Calabi–Yau threefolds of type K and have studied some basic properties thereof. In the present paper, we continue the study, investigating them from the viewpoint of mirror symmetry. It is shown that mirror symmetry relies on duality of certain sublattices in the second cohomology of the K3 surface appearing in the minimal splitting covering. The duality may be thought of as a version of the lattice duality of the anti-symplectic involution on K3 surfaces discovered by Nikulin [23]. Based on the duality, we obtain several results parallel to what is known for Borcea–Voisin threefolds. Along the way, we also investigate the Brauer groups of Calabi–Yau threefolds of type K.

We continue our study of the chemical (graph) distance inside large critical percolation clusters in dimension two. We prove new estimates, which involve the three-arm probability, for the point-to-surface and point-to-point distances. We show that the point-to-point distance in Z 2 between two points in the same critical percolation cluster has infinite second moment. We also give quantitative versions of our previous results comparing the length of the shortest crossing to that of the lowest crossing of a box.

We study chiral algebras associated with Argyres-Douglas theories engineered from M5 brane. For the theory engineered using 6d (2,0) type J theory on a sphere with a single irregular singularity (without mass parameter), its chiral algebra is the minimal model of W algebra of J type. For the theory engineered using an irregular singularity and a regular full singularity, its chiral algebra is the affine Kac-Moody algebra of J type. We can obtain the Schur index of these theories by computing the vacua character of the corresponding chiral algebra.

In this paper, we show the equivalence between two seemingly distinct 2d TQFTs: one comes from the "Coulomb branch index" of the class S theory T[Σ,G] on L(k,1)×S1, the other is the LG "equivariant Verlinde formula", or equivalently partition function of LGℂ complex Chern-Simons theory on Σ×S1. We first derive this equivalence using the M-theory geometry and show that the gauge groups appearing on the two sides are naturally G and its Langlands dual LG. When G is not simply-connected, we provide a recipe of computing the index of T[Σ,G] as summation over indices of T[Σ,G̃ ] with non-trivial background 't Hooft fluxes, where G̃ is the simply-connected group with the same Lie algebra. Then we check explicitly this relation between the Coulomb index and the equivariant Verlinde formula for G=SU(2) or SO(3). In the end, as an application of this newly found relation, we consider the more general case where G is SU(N) or PSU(N) and show that equivariant Verlinde algebra can be derived using field theory via (generalized) Argyres-Seiberg duality. We also attach a Mathematica notebook that can be used to compute the SU(3) equivariant Verlinde coefficients.

The absorption rate of low-energy, or soft, electromagnetic radiation by spherically symmetric black holes in arbitrary dimensions is shown to be fixed by conservation of energy and large gauge transformations. We interpret this result as the explicit realization of the Hawking-Perry-Strominger Ward identity for large gauge transformations in the background of a non-evaporating black hole. Along the way we rederive and extend previous analytic results regarding the absorption rate for the minimal scalar and the photon.